Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 42

Chapter 3, Problem 42

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium's average velocity for the entire trip?

Verified step by step guidance

Step 1: Understand the concept of average velocity. Average velocity is defined as the displacement vector divided by the total time taken. Displacement is the straight-line distance and direction from the initial position to the final position.

Step 2: Identify the initial and final positions of the bacterium from the graph. The bacterium starts at point A (O) with coordinates (0, 0) and ends at point E (S) with coordinates (30 cm, 0 cm).

Step 3: Calculate the displacement vector. The displacement is the difference between the final position and the initial position. In this case, the displacement vector is Δx = 30 cm - 0 cm = 30 cm and Δy = 0 cm - 0 cm = 0 cm. Thus, the displacement vector is (30 cm, 0 cm).

Step 4: Determine the magnitude of the displacement. The magnitude of the displacement is calculated using the formula: |Δr| = √((Δx)^2 + (Δy)^2). Substituting the values, |Δr| = √((30)^2 + (0)^2) cm.

Step 5: Calculate the direction of the displacement. The direction is given by the angle θ with respect to the positive x-axis. Since Δy = 0, the displacement is entirely along the x-axis, and θ = 0 degrees. Combine this with the constant speed of 20 μm/s to find the average velocity vector.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity, meaning it has both magnitude and direction. In the context of the E. coli's movement, calculating average velocity involves determining the straight-line distance from the starting point to the endpoint and the time taken for the entire journey.

Displacement

Displacement refers to the change in position of an object and is a vector quantity. It is calculated as the difference between the final and initial positions, taking into account the direction. For the E. coli, displacement can be found by measuring the straight-line distance between the starting point (point A) and the endpoint (point E) on the graph.

Trajectory

Trajectory is the path that an object follows as it moves through space. In the case of the E. coli, its zig-zag path represents its trajectory in the medium. Understanding the trajectory is essential for analyzing the motion and calculating average velocity, as it provides insight into the overall movement pattern of the bacterium.

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Textbook Question

A crate, seen from above, is pulled with three ropes that have the tensions shown in FIGURE P3.44. Tension is a vector directed along the rope, measured in newtons (abbreviated N). Suppose the three ropes are replaced with a single rope that has exactly the same effect on the crate. What is the tension in this rope? Write your answer in component form using unit vectors.

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Textbook Question

Tom is climbing a 3.0-m-long ladder that leans against a vertical wall, contacting the wall 2.5 m above the ground. His weight of 680 N is a vector pointing vertically downward. (Weight is measured in newtons, abbreviated N.) What are the components of Tom's weight parallel and perpendicular to the ladder?

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Textbook Question

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis.

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Textbook Question

The treasure map in FIGURE P3.41 gives the following directions to the buried treasure: 'Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig.' But when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60° east of north. After walking 300 paces you see an opening through the woods. In which direction should you walk, as an angle west of north, and how far, to reach the treasure?

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Textbook Question

Four forces are exerted on the object shown in FIGURE $P 3.45$ . (Forces are measured in newtons, abbreviated $N$ .) The net force on the object is ${\vec{F}}_{net} = {\vec{F}}_{1} + {\vec{F}}_{2} + {\vec{F}}_{3} + {\vec{F}}_{4} = 4.0 \hat{i} N$ . What are (a) $F ⃗ sub 3$ and (b) $F ⃗ sub 4$ ? Give your answers in component form.

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Textbook Question

Your neighbor Paul has rented a truck with a loading ramp. The ramp is tilted upward at 25°, and Paul is pulling a large crate up the ramp with a rope that angles 10° above the ramp. If Paul pulls with a force of 550 N, what are the horizontal and vertical components of his force? (Force is measured in newtons, abbreviated N.)

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